Given this apparatus, it makes sense to try to directly find the appropriate values of the derivatives at the nodes directly from the given function values and derivative values at the nodes; that is, by finding the differentiation matrix.
Rather than give the derivation (a complete one can be found in chapter 11 of [1]}) we point to both Maple code and Matlab code that implements those formulas, at http://www.nfillion.com/coderepository/Graduate_Introduction_to_Numerical_Methods/ in BHIP.mpl and genbarywts.m, respectively. Evaluation in Matlab can be done with the code hermiteeval.m. We give an example below.
